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Homogeneous S-Lemma and its application to asymptotic stability of a class of switched nonlinear systems

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2 Author(s)
Zhang Kuize ; College of Automation, Harbin Engineering University, Harbin 150001, P.R. China ; Zhang Lijun

This paper extends the strict S-Lemma proposed by Yakubovich and uses the improved strict S-Lemma to investigate the asymptotic stability of a class of switched nonlinear systems. First, the strict S-Lemma is extended from quadratic forms to homogeneous polynomials, where the improved S-Lemma is named the strict homogeneous S-Lemma (short for the SHS-Lemma). Then by utilizing the SHS-Lemma, it is proved that a switched nonlinear polynomial system with two sub-systems admits a Lyapunov function with homogeneous derivative (short for LFHD) if and only if it has a convex combination of the vector fields of its two sub-systems that admits a LFHD. Furthermore, it is shown that the “if” part of the former property still holds for switched polynomial systems with three or more sub-systems but the “only if” part does not even for switched linear systems.

Published in:

Control Conference (CCC), 2012 31st Chinese

Date of Conference:

25-27 July 2012